Find the LCM of 36, 84 and 90 ?

Answer

To find the LCM (Least Common Multiple) of the numbers 36, 84, and 90, we will follow these steps:

### Step 1: Prime Factorization
First, we need to factor each number into its prime factors.

- **For 36**:
- 36 can be divided by 2:
- \( 36 \div 2 = 18 \)
- 18 can be divided by 2:
- \( 18 \div 2 = 9 \)
- 9 can be divided by 3:
- \( 9 \div 3 = 3 \)
- 3 is a prime number.

So, the prime factorization of 36 is:
\[ 36 = 2^2 \times 3^2 \]

- **For 84**:
- 84 can be divided by 2:
- \( 84 \div 2 = 42 \)
- 42 can be divided by 2:
- \( 42 \div 2 = 21 \)
- 21 can be divided by 3:
- \( 21 \div 3 = 7 \)
- 7 is a prime number.

So, the prime factorization of 84 is:
\[ 84 = 2^2 \times 3^1 \times 7^1 \]

- **For 90**:
- 90 can be divided by 2:
- \( 90 \div 2 = 45 \)
- 45 can be divided by 3:
- \( 45 \div 3 = 15 \)
- 15 can be divided by 3:
- \( 15 \div 3 = 5 \)
- 5 is a prime number.

So, the prime factorization of 90 is:
\[ 90 = 2^1 \times 3^2 \times 5^1 \]

### Step 2: Identify the Highest Powers
Next, we will identify the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^2\) (from 36 and 84).
- For \(3\): The highest power is \(3^2\) (from 36 and 90).
- For \(5\): The highest power is \(5^1\) (from 90).
- For \(7\): The highest power is \(7^1\) (from 84).

### Step 3: Calculate the LCM
Now, we will multiply these highest powers together to find the LCM:

\[ \text{LCM} = 2^2 \times 3^2 \times 5^1 \times 7^1 \]

Calculating this step-by-step:

1. Calculate \(2^2 = 4\)
2. Calculate \(3^2 = 9\)
3. Calculate \(5^1 = 5\)
4. Calculate \(7^1 = 7\)

Now multiply these results together:

\[ \text{LCM} = 4 \times 9 \times 5 \times 7 \]

Calculating step-by-step:

1. \(4 \times 9 = 36\)
2. \(36 \times 5 = 180\)
3. \(180 \times 7 = 1260\)

Thus, the LCM of 36, 84, and 90 is:
\[ \text{LCM} = 1260 \]

### Final Answer
The LCM of 36, 84, and 90 is **1260**.

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